Ratio Method of Estimation in the Presence of Measurement Errors

Jour.lnd. Soc. Ag. Stalistics 50(2),1997: 150-155

Ratio Method of Estimation in the Presence of Measurement Errors Shalabb University of Jammu, Jammu-180004 (Received: May, 1996) SUMMARY In the present article, the effect of measurement errors on the ratio estimation technique of the population mean is examined. Key words: Ratio method, Measurement error.

J. illlroduClioll

In survey sampling, tbe properties of estimators based on data originating under various kinds of sampling schemes and various ways of estimation procedure are generally analyzed under the supposition that observations have been recorded without any error. Such a supposition may not be tenable in actual practice and the data may (:ontaiu observational or measurement errors due to various reasons; see, e.g., Cochran [1] and Sukhatme el at. [2]. An important source of measurement errors in survey data is the nature of variables. TIle nature of the variable arising from the definition may often be such that exact measurements on it are not available. This may happen chiefly due to three reasons. First is that the variable is clearly defined but it is hard to take correct observations at least with the currently available techniques or because of other types of practical difficulties. Consequently, imperfect measurements are obtained sllch as in the case of montldy expenditure on food items in a household or the level of blood sugar in a human being. Second is that the variable is conceptually well defined but observations can be obtained only on some closely related substitutes known as proxies or surrogates. A simple example is the measurement of economic status of a persoll or the level of cducation 011 which true observations cannot be obtained. One may, for instance, use tile lllunber of years of schooling for the level of education, and it may serve as a reasonably good approximation. Third is that the variable is fully comprehensible and well understood but it is 110t intrinsically defincd and properly quantified. However, it is known to be closely associated with a number of factors. Simple examples of such variables are intelligence. specific

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RATlO METHOD OF ESTIMATION

abilities. utility. aggressiveness. etc. Any theoretical characteristic having at least one of these features is popularly labelled as unobservable or latent variable. True observations on such characteristics are genemlly unavailable. The reported observations are thus contaminated by measurement errors.

III this article. we consider the estimation of population mean arising from a popular method of estimation, viz., ratio meUlOd and analyze its properties in the presence of measurement errors.

2. Main Results Suppose that we are given a set of It paired observations obtained through simple random sampling procedure. on two characteristics X and Y. It is asslIDled that Xi and Yi for the jill sampling unit are recorded instead of true values Xi and

Vi'

The observational or measurement errors are defined as Ui = (Yj- Vi)

(2.1)

vi = (Xi -Xi)

(2.2)

which are assmned to be stochastic with mean 0 but possibly different variances 2 2 0 u and 0 v' For the sake of simplicity in exposition, we assume that ui's and v/s are uncorrelated although Xi's and Yi' S are correlated. Such a specification can be, however, relaxed at Ule cost of some algebraic complexity. We also aSSlUue that finite population correction can be ignored.

Let Ule population means of X and Y characteristic be f.1x and J.ly and population variances

oi and o~. Furtiler, let p be the population correlation

coefficient between X and Y. For tbe estimation of population mean lly. Ule traditional unbiased estimator is tile San11)le mean y but it does not utilize the sample information Oil X characteristic. One popular way to incorporate it is tbe ratio meUlod. Assuming Ulaf Ilx is known and is different from zero, this method yields the following estimator of Ily

:

tR

where

=

(f )IlX

x denotes the mean of sample observations on X.

(2.3)

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JOURNAL OF 11IE INDIAN SOCIErr OF AGRICULTURAL STATISflCS

In order to study dIe efficiency properties of dIe estimators

y and

tR in

the presence of measurement errors, we frrst introduce the following notations:

ex

Ox = -,

Jlx

= n_ 112 .L.Vj, 'C"

Wv

Wx

= n- 112I:(X j -

Jlx)

It is easy to see dlat (2.4)

-- n- I12 (wy +w) u whence we observe tbat

V(y) Next,

w~

y is

unbiased willi variance

= -~( 1+- o~J 2 11

0

(2.5)

Y

can express

Wy+wu

Jly +

11112

(2.6)

= -nal12- -ab + 0 (n-312) n P where (2.7)

153

RATIO ME11IOD OF ESTIMATTON

Thus the bias and mean squared error of la upto order O(n- 1) are given

by (2.8) ~y

°v2]

= - Cx(Cx -pCY)+2"

n

[

~x

(2.9)

~

=-[ 11

1--(2p--)]+- Ou+(-] 1 Cx Cy

Cx Cy

1

2

n[

~y ~x

2

2 0v

Looking at the expressions presented above, we observe that the measurement errors have no influence at all on the unbiasedness of y but for the ratio estimator ~, only the errors in auxiliary characteristic X affect the bias, at least to the order of our approximation. Examining the expressions (2.5) and (2.9), we observe that sampling variability in each case increases when measurement errors are present. It is interesting to note that the increase in variability attributable to measurement errors is small in case of y when compared WiUl that of tit. Next, we find that Ule estimator ~ is superior to criterion of mean squared error upto order 0(n-

1

)

y with

when

0; 1

x

respect to the

C ( 1 + 2" if ~x and ~y have same signs p > 2C y

(2.10)

Ox

x

fl';

C ( 1 + 2" p < - 2C y

Ox

1

if ~x and ~y have opposite sigm(2.1l)

In a particular case where Cx and C y are identical in magnitudes, these conditions reduce to the following:

~

p > ( 1+ :

--

1~x if

and

~y bave same siglls

(2.12)

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JOURNAL OF mE INDIAN SOCIElY OF AGRICULTURAL STATISTICS

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